Projective representation theory of quaternion group

This article gives specific information, namely, projective representation theory, about a particular group, namely: quaternion group.
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This article describes the projective representation theory of the quaternion group, which we call $G$, in characteristic zero.

First, note that the Schur multiplier $H^{2}(G,\mathbb {C} ^{\ast })$ is isomorphic to trivial group. For more, see group cohomology of quaternion group.

Therefore, the only kinds of projective representations are those arising from linear representations, i.e., those for the trivial cohomology class. Thus, the irreducible projective representations are the same thing as the equivalence classes of irreducible linear representations under the multiplicative action of one-dimensional representations.

Here is a list of the irreducible projective representations and linear representations that give rise to them:

Representation	Degree	Number of linear representation that give rise to it	List of linear representations
trivial	1	4	trivial representation and the three sign representations with kernels $\langle i\rangle$, $\langle j\rangle$, and $\langle k\rangle$ respectively
irreducible two-dimensional	2	1	faithful irreducible representation of quaternion group